AMC8 2024

AMC8 2024 · Q24

AMC8 2024 · Q24. It mainly tests Triangles (properties), Area & perimeter.

Jean has made a piece of stained glass art in the shape of two mountains, as shown in the figure below. One mountain peak is $8$ feet high while the other peak is $12$ feet high. Each peak forms a $90^\circ$ angle, and the straight sides form a $45^\circ$ angle with the ground. The artwork has an area of $183$ square feet. The sides of the mountain meet at an intersection point near the center of the artwork, $h$ feet above the ground. What is the value of $h?$

Jean 制作了一件彩色玻璃艺术品，形状如两座山，如下图所示。一座山峰高 $8$ 英尺，另一座高 $12$ 英尺。每座山峰形成 $90^\circ$ 角，直边与地面形成 $45^\circ$ 角。艺术品面积为 $183$ 平方英尺。两山侧边在艺术品中心附近相交，离地面 $h$ 英尺。$h$ 的值为多少？

(A) \ 4 \ 4

(B) \ 5 \ 5

(D) \ 6 \ 6

(E) \ 5\sqrt{2} \ 5\sqrt{2}

Answer

Correct choice: (B)

正确答案：（B）

Solution

Extend the "inner part" of the mountain so that the image is two right triangles that overlap in a third right triangle as shown. The side length of the largest right triangle is $12\sqrt{2},$ which means its area is $144.$ Similarly, the area of the second largest right triangle is $64$ (the side length is $8\sqrt{2}$), and the area of the overlap is $h^2$ (the side length is $h\sqrt{2}$). Because the right triangles have a side ratio of 1:1:$\sqrt{2}$.Thus, \[144+64-h^2=183,\] which means that the answer is $\boxed{\mathbf{(B)}\text{ 5}}.$

将山的“内侧”部分延伸，使图像成为两个重叠的直角三角形，如图所示。最大直角三角形的斜边长为 $12\sqrt{2}$，面积为 $144$。第二大三角形斜边长 $8\sqrt{2}$，面积为 $64$，重叠部分为第三个直角三角形，面积 $h^2$（斜边 $h\sqrt{2}$）。因为这些直角三角形边比为 $1:1:\sqrt{2}$。因此，$144+64-h^2=183$，解得 $h=5$，答案为 $\boxed{\mathbf{(B)}\text{ 5}}$。

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